The Infinite limit of random permutations avoiding patterns of length three. (14th January 2020) - Pinsky, Ross G.
Abstract: For $$τ \in {S_3}$$, let $$μ _n^τ $$ denote the uniformly random probability measure on the set of $$τ $$ -avoiding permutations in $${S_n}$$ . Let $${\mathbb {N}^*} = {\mathbb {N}} \cup \{ ∞ \} $$ with an appropriate metric and denote by $$S({\mathbb{N}}, {\mathbb{N}^*})$$ the compact metric space consisting of functions $$σ {\rm{ = }}\{ {σ _i}\} _{i = 1}^∞ {\rm{ }}$$ from $$\mathbb {N}$$ to $${\mathbb {N}^ * }$$ which are injections when restricted to $${σ ^{ - 1}}(\mathbb {N})$$ ; that is, if $${σ _i}{\rm{ = }}{σ _j}$$, $$i \ne j$$, then $${σ _i} = ∞ $$ . Extending permutations $$σ \in {S_n}$$ by defining $${σ _j} = j$$, for $$j \gt n$$, we have $${S_n} \subset S({\mathbb{N}}, {{\mathbb{N}}^*})$$ . For each $$τ \in {S_3}$$, we study the limiting behaviour of the measures $$\{ μ _n^τ \} _{n = 1}^∞ $$ on $$S({\mathbb{N}}, {\mathbb{N}^*})$$ . We obtain partial results for the permutation $$τ = 321$$ and complete results for the other five permutations $$τ \in {S_3}$$ .